A009759 Expansion of (3 - 21*x + 4*x^2)/((x-1)*(x^2 - 6*x + 1)).

-3, 0, 17, 116, 693, 4056, 23657, 137900, 803757, 4684656, 27304193, 159140516, 927538917, 5406093000, 31509019097, 183648021596, 1070379110493, 6238626641376, 36361380737777, 211929657785300, 1235216565974037

Offset: 0

Author

N. J. A. Sloane, Nick Baxter (nick(AT)visigenic.com)

Keywords

• sign

Comments

Numbers k such that 2*k^2 + 14*k + 25 is a square. - James R. Buddenhagen, Jul 19 2008

If k satisfies the previous condition, i.e., P(k) = 2*k^2 + 14*k + 25 is a square, then there exists an integer m such that (k+3)^2 + (k+4)^2 = m^2; finding such numbers k is equivalent to finding Pythagorean triples (X,X+1,Z) with X in A001652 (1st formula). For k >= 0, P(k) = A001844(k+3), and the successive values of P(k) that are squares are in A008844. - Bernard Schott, Mar 24 2019

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, New-York, 1964, pp. 122-124.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A001652 (X values), A001653 (Z values), A001844, A008844.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (3-21*x+4*x^2)/((x-1)*(x^2-6*x+1))) ) // G. C. Greubel, Feb 12 2018

Mathematica

CoefficientList[Series[(3-21x+4x^2)/((x-1)(x^2-6x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{7,-7,1},{-3,0,17},30] (* Harvey P. Dale, Dec 12 2016 *)

PARI

my(x='x+O('x^30)); Vec((3-21*x+4*x^2)/((x-1)*(x^2-6*x+1))) \\ G. C. Greubel, Feb 12 2018

Sage

((3-21*x+4*x^2)/((x-1)*(x^2-6*x+1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 04 2019

Formula

a(n) = A001652(n) - 3.

a(n) = ( (1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1) - 14 )/4.

a(n) - a(n-1) = A001541(n), n > 0. - R. J. Mathar, Apr 23 2009

a(n) = (Q(2*n+1) -14)/4 = (4*P(n)*P(n+1) + (-1)^n - 7)/2, where P(n) = A000129(n) (Pell) and Q(n) = A002203(n) (Pell-Lucas). - G. C. Greubel, Apr 04 2019

Extensions

G.f. and Binet formula corrected by R. J. Mathar, Aug 24 2016